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What is a Pairwise Comparison? Pairwise comparisons start with a given set of objects and preferences between the objects. Based on these preferences between pairs of objects, various methods can be used to determine which of the objects is “most” preferred. All methods of comparisons rank the objects by assigning a ranking number or strength to each object. In the methods we are considering, strengths are assigned using advance tools involving the probability that an object will be preferred to another in a future comparison. Primary Question Many methods of ranking exist each with varying criteria and conditions on the objects being compared so that unique rankings are produced. Three methods of ranking include the Zermelo approach [2] (also referred to as the Bradley-Terry method from psychologist Bradley and Terry who studied pairwise comparisons) [1], the Jech approach [3], and the uniform model [4]. Stob explains that the Zermelo and Jech methods will usually result in the same rankings. But Stob also notes that the uniqueness conditions differ greatly between the two methods [5]. For my research, I will explore the differences between the three methods of ranking in hopes of creating bounds so that the three methods produce the same ranking. What is a bound for the differences in strengths from various pairwise comparison models? Uniqueness Condition for the Zermelo Approach Studied by Ford in 1957 [2] Restrictions Bi-partition of object can be created so that an object in the first partition is preferred to another object in the second partition. Similarly, an object in partition 2 is preferred to an object in partition 1. Path of wins can be created from any two objects in sets. For example, we are trying to compare teams A and D from the above graph. The path of wins would look like We follow the directed arrows from object A to object D, knowing A is preferred to C, C is preferred to B, and B is preferred to D. Uniqueness Condition for the Jech Approach Studied by Jech in 1983 [3] Restrictions Cycles of probabilities with the same teams are equal in value P ab P bc P ca = P ac P cb P ba This restriction gives us that transitivity between teams must hold. This means that if object A is preferred to B and B to C, then A should be preferred to C. Important Concepts Graph Theory- Uses a system of directed graphs to illustrate preference in comparisons. Arrows point from the preferred in the object comparison. Matrices – Used to store information for sets of data abc a021 b201 c110 Uniqueness Conditions- Given any pairwise comparison method, uniqueness conditions are a set of restrictions placed on the data so that a single ranking is produced. For example, if strengths were assigned to teams based on number of wins, there is only one ranking for the teams that meets this criteria. A - 3 winsC B - 2 winsRanking A C - 4 winsB References [1] Bradley, Ralph, and Milton Terry. (1952). Rank Analysis of Incomplete Block Designs. Biometrika, 39(3/4), [2] Ford, L.R. (1957). Solution of a Ranking Problem from Binary Comparison. The American Mathematical Monthly, 64(8), [3] Jech, Thomas. (1983). The Ranking of Incomplete Tournaments: A Mathematician’s Guide to Popular Sports. The American MathematicalMonthly, 90(4), [4] Smith, J.H. (1956). Adjusting Baseball Standings for Strength of Teams Played. American Statistician, 10, [5] Stob, Michael. (1984). A Supplement to "A Mathematician's Guide to Popular Sports". The American Mathematical Monthly, 91(5), usageorge.com A B C D A B C B A C ABCD Why do we care? Suppose we are looking at a sports conference where the season is unfinished, meaning not all teams have played each other an equal number of times. If each game is a pairwise comparison, we can use our methods to determine a conference champion. Suppose you are looking to buy a house. Many factors go into making this decision, but suppose you want to know which few factors are most important to consider. Using weighted pairwise comparisons, we can determine which factors are most important when making the decision. Ranking systems have application in many fields including: Donor Lists Search Engine Results Taste Testing AB

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